# Talk:Hexagon

## Why is the beehive honeycomb hexagonal?

...Muazzes ...

"Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane (three hexagons meeting at every vertex), and so are useful for constructing tessellations."

      Up to here it is ok, but see how it goes -->


"The cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials."

I propose this second line goes away... —Preceding unsigned comment added by Muazzes (talkcontribs) 12:26, 3 August 2008 (UTC)

## picture is good now

...Anthere ...

...sannse 21:26 May 13, 2003 (UTC)

Yeah, that is good. Nope, I didnot need any refresh :-) ant A regular hexagon is constructible with compass and straightedge. The following is a step-by-step animated method of this.

## picture of honeycomb ?

I think we need a picture of a honeycomb to illustrate "hexagonal cells".

The Main Page currently mentions "the ideal cellular network has evenly distributed hexagonal cells"

-- but clicking on hexagonal just redirects to hexagon, which doesn't say anything about hexagonal cells (hexagonal arrays). It only talks about single hexagons.

-- DavidCary 03:31, 20 Jun 2004 (UTC)

## number of regions

Do any of you Wikipedians know the number of regions a hexagon with all of its diagonals shown determines?? 66.245.12.170 23:05, 25 May 2004 (UTC)

A regular hexagon has 25. A general hexagon, assumed to have no more than two diagonals intersecting at any point (other than a vertex) has 26. -- Smjg 11:08, 26 May 2004 (UTC)

See Talk:Diagonal for more details.

## "Sexagon"

Is "sexagon" really a good term?? I say no. "Hexagon" is completely Greek. "Sex" is a Latin prefix. If the word were Latin, it would be "sexangle". See Greek numerical prefixes and Latin numerical prefixes. 66.245.11.175 14:05, 15 Jun 2004 (UTC) indeed m.e. 12:03, 18 Jun 2004 (UTC)

I don't think I've ever heard it called a sexagon, though the equally mixed-up nonagon for enneagon is common, and I've also heard of septagon instead of heptagon. So if the term's to be mentioned, I suppose it ought to have some wording discouraging its use.... -- Smjg 16:23, 18 Jun 2004 (UTC)
While you're at it, why not discourage television too? : ) Doops 05:11, 20 Jul 2004 (UTC)
"Television" does not have a Greek or Latin numerical prefix, so what's so special about it?? 66.245.103.69 01:08, 21 Aug 2004 (UTC)

Yes, but tele is Greek, and vision is Latin... which gets right to the heart of your litanies on these topics. It doesn't matter whether you think sexagon is a "good term" or not. I have any number of late XIX and early XX century math texts that have it, along with septagon, octagon, nonagon, and also undecagon and duodecagon. Do you propose we go back in time and remove those? You're sticking in a reasonable point, about the foolish inconsistency, but it's well established in literature that the Latinate use of the terms was always mixed with Greek. Your purported linguistic scholarship of insisting "sexangle" would be a Latin word is extremely incorrect. "Angle" is an English word derived from Latin "angulus" via Norman French thru Middle English. Latin didn't use "sexangulus" in any event. Furthermore, the Greek gon component is related to Latin genu and Modern English knee. Latin's use of the e-grade form beside Greek's o-grade form, didn't preclude Latin speakers from having too much trouble taking gon's meaning correctly. --Sturmde 18:04, 14 July 2005 (UTC)

I don't much care for the Latin and Greek prefixes and suffixes. I speak English, and the accepted English words are Hexagon, Nonagon and Television. Not enough people care enough about the history of words to change that. 128.240.229.66 20:02, 10 August 2006 (UTC)
I prefer "enneagon", but then either people don't understand me or correct it to "nonagon". Professor M. Fiendish, Esq. 01:46, 9 September 2009 (UTC)

## Comparing to a rectangle?

Current version: “The Area of a rectangle. You can compare the area of a hexagon to a rectangles.If you make triangles from the hexagon.You can make it into a triangle by switching the triangles around.”

Editing errors not withstanding, I don’t know if he/she meant rectangle or triangle.

If you partition the hexagon into equilateral triangles you can rearrange the 6 of them to make a parallelogram but never a rectangle because all the angles are 60 degrees and could never be combined to be 90. Or assuming he/she meant triangle, it takes 9 equilateral triangles to make a triangle. Unless he/she means something else by that like making right angle cuts in the hexagon and fashioning right triangles and perhaps a rectangle in the middle. But you would have to butcher it and it would no longer yield an intuitive lesson. Perhaps he or she is referring to an inscribed or circumscribed rectangle? Is the originator of this section a native English speaker?

## Minimal diameter formula

Can I change the formula for the minimal diameter? I read it as 'root (3a)' instead of '(root 3)a' and it confused me for a few minutes. Would 'a root 3' be better? Tomid 20:11, 10 August 2006 (UTC)

## Hexagon and square

Is it true that a square whose sides each measure ten centimeters can completely fit inside of a regular hexagon whose sides each measure ten centimeters ?

I don't think so. I'm no expert but I know that the area of such a square is 100 cm2, and I think the area of the hexagon is ${\displaystyle 150{\sqrt {3}}}$, which clearly isn't possible. However, I think there is some true theory involving a cube of 10 cm sides and a hexagonal hole with 10cm sides, but I don't know much about it. Maybe you could try the reference desk? Hyenaste (tell) 04:45, 28 August 2006 (UTC)
By my calculation, it's perfectly possible for the area of a regular hexagon of side 10 cm to be ${\displaystyle 150{\sqrt {3}}}$ cm2. What do you mean exactly? -- Smjg 23:26, 29 August 2006 (UTC)
I mean it is not possible for shape with an area of ${\displaystyle 150{\sqrt {3}}}$ cm2 to fit inside an area of 100cm2. Hyenaste (tell) 01:53, 30 August 2006 (UTC)
You've got it the wrong way round. The rest of us are talking about a square of side 10cm, area 100cm2, fitting inside a hexagon of side 10cm, area ${\displaystyle 150{\sqrt {3}}}$ cm2. Not of the hexagon fitting inside the square. -- Smjg 14:47, 16 September 2006 (UTC)
Yeah, I realise that now. I read his question, thought about the "hole-through-the-cube" I mentioned, and assumed he was asking a different question. Oh well. Hyenaste (tell) 23:03, 16 September 2006 (UTC)
Easily so, I would say. Imagine a hexagon with side length 10cm. Take one of those sides and draw a 10cm x 10cm square from it, inside the hexagon. The hexagon is much bigger. Tomid 15:05, 31 August 2006 (UTC)

Cleaned up some vandalism.

This is the most retardedly obvious question I have ever seen. Just line up one edge of the square with one of the hexagon. Itscalledhexagon (talk) 14:02, 20 October 2011 (UTC)

Confused...

A Circle has a total of 360 degrees. According to the Article, each interior angle has 120 degrees. The sum of the interior angles would be 720 degrees. If the interior angle was reduced to 60 degrees, then a sum of 360 degrees would be achieved.

paul d.

I'm no geometrician, but I think you're confused about interior angles. The interior angles of a polygon do not have to equal 360°. If, like you suggested, each angle was compressed to 60°, the shape would become two equilateral triangles, the second a trace of the first. Hyenaste (tell) 23:03, 16 September 2006 (UTC)

If one were to measure the angle from a point (A) in the exact center of a perfect hexagon, to a line that intersects the angles of the hexagon (AB and AC) making a triangle; how would one figure out that new angle? I cannot seem to find any information on it. Thank you for your help ISO 1806: 2007-08-12 T05:40 Z-7 76.170.117.217 12:39, 12 August 2007 (UTC)

I think you are describing the angle subtended at the centre by a side. If so, then this is indeed 60 degrees for a regular hexagon. For a regular n-gon, this angle is always 360 divided by n degrees. Dbfirs 17:42, 2 May 2011 (UTC)

## Pictures of Tesselations

Apart from the colours, how do the first and third pictured tesselations differ? The third would seem to be the same as the first, only tilted a bit. Rojomoke (talk) 14:01, 23 October 2008 (UTC)

All of the top three hexagonal tilings are geometrically identical, except for different symmetry as represented by the colors. Tom Ruen (talk) 19:41, 23 October 2008 (UTC)
The three are exactly the same tessellation. The first is the base pattern, and the second and third are just two ways of colouring this pattern. Could do with better wording. I'll see what I can come up with. -- Smjg (talk) 09:10, 11 February 2009 (UTC)

opposite of a regualr hexagon are what to each other —Preceding unsigned comment added by 75.85.250.91 (talk) 02:23, 19 February 2009 (UTC)

## Free Market simulation article in the 1970s

In the late 1970s, there was an article in Scientific American magazine about market dynamics in Imperial China in the Middle Ages. The idea of using a hexagonal array of markets and sub-markets arises from the observation that the forces of supply and demand tend to oscillate, or alternate, much like a teeter totter rises and falls, and far from going out of balance, actually tends to arrange itself in a dynamic balance of sorts.

The main article could be improved if there were a reference to the magazine article about market forces arranging themselves into hexagons. 216.99.201.35 (talk) 20:34, 19 June 2009 (UTC)

uh .. Did the article say that "market forces arrange themselves into hexagons" or merely use hexagons for the same reason that tabletop wargames often do? —Tamfang (talk) 18:11, 30 November 2010 (UTC)

## Huh?

Why does the caption to the animated gif say:

"A regular hexagon is constructible with compass and straightedge. The following is a step-by-step animated method of this, given by Euclid's Elements, movie IV, Proposition 15." —Preceding unsigned comment added by 116.14.20.208 (talk) 04:44, 18 July 2009 (UTC)

Fixed. Professor M. Fiendish, Esq. 01:47, 9 September 2009 (UTC)

## Use of hexagons in construction

Wouldn't a section or a new article regarding "Use of Hexagon's in construction" be useful ? Appearantly, bees have chosen the hexagon since it requires less material than making ie the combs from triangles. Thus, it is still a very good construction shape ie when building storage cells (ie for food, ...) 91.182.241.204 (talk) 13:20, 30 November 2010 (UTC)

Well, the article already contains this sentence: The cells of a beehive honeycomb are hexagonal [...] because the shape makes efficient use of space and building materials.Tamfang (talk) 18:09, 30 November 2010 (UTC)

## triambus

Template:User-multi added this sentence to the Regular hexagon section:

It is not usually considered a triambus, although it is equilateral.

I had not known the word triambus before. The link to equilateral polygon implies that a triambus is any hexagon with equal sides; but the name suggests that alternately-equal angles are also a defining feature. Which is it?

Let's have a new section about triambs. —Tamfang (talk) 18:39, 1 May 2011 (UTC)

I can't find much about them except for thisthat suggests just normal equilateral hexagons. Neither Wiktionary nor the big Oxford English Dictionary defines any of "triamb", "triambus" and "triambic". Are they notable enough to mention, and, if so, what are they? Dbfirs 20:33, 1 May 2011 (UTC)
From usage, a triamb(us) appears to be a hexagon with equal sides and threefold rotational symmetry. A regular hexagon is to triambs as a square is to rhombs. —Tamfang (talk) 23:44, 1 May 2011 (UTC)
Thanks. I can picture them now. (e.g. take two squares and cut each across a diagonal then take three of the four right-angled pieces and fit them together to form a hexagon with alternating right-angles and 150 degree angles.) Would someone like to upload an image? Dbfirs 06:57, 2 May 2011 (UTC)
Truncated tetrahedron (a.k.a. Triambic-Triangular Octahedron). Truncated octahedron (a.k.a. Triambic-Square Tetradecahedron). Truncated icosahedron (a.k.a. Triambic-Pentagonal Triacontahdihedron).KirbyRider (talk) 08:14, 2 May 2011 (UTC)
... so is a "triamb" any 2-D or 3-D shape with rotational symmetry of order 3? If so, is an equilateral triangle the simplest triamb? Dbfirs 17:35, 2 May 2011 (UTC)
I've not seen these names used for convex polyhedra, but seems to be used in the uniform star polyhedral duals like medial triambic icosahedron, (and similar great triambic icosahedron), names created by Norman Johnson I expect! If Tamfang's definition is correct, this term doesn't rightfully belong as usage in regular hexagons - that would be like calling a square a rhombus, true but under-qualified. Tom Ruen (talk) 00:56, 2 May 2011 (UTC)
I suppose one could call them semi-regular hexagons? The adjective triambic possibly deserves a Wiktionary entry since it seems to be used in the works of more than one author. Is the same true of triamb and triambus? Dbfirs 06:57, 2 May 2011 (UTC)
Semi-regular isn't the right word. Semiregular polyhedra are vertex transitive. These would be isotoxal figures (edge-transitive), so it would be an isotoxal hexagon. Tom Ruen (talk) 18:52, 2 May 2011 (UTC)
Thanks, I wasn't aware that mathematicians had chosen to use "semi-regular" for only half of it's possible meaning. It's not a precise description anyway -- your precise terms are much better, but might be slightly too technical for a basic article on a 2-D figure because they are normally applied to polyhedra. Do you think we should add a section on equilateral hexagons that are not equiangular but have some rotational symmetry? (and, I suppose, on equiangular hexagons that are not equilateral but have a line of symmetry) -- or are they too trivial to be worth mentioning? Dbfirs 07:36, 3 May 2011 (UTC)
A triambus is a hexagon which is equilateral but not equiangular (a sort of hexagonal rhombus), while a ditrigon is a hexagon with alternating edge lengths and is equiangular (a sort of hexagonal rectangle). These two can be extended to any number of sides: an octambus would be an equilateral but not equiangular hexadecagon, and a dipentagon would be an equiangular decagon with alternating edge lengths. (The di-n-gon can also be extended to fractional n, like the dipentagram or distellagon when n = Template:Frac.) Double sharp (talk) 09:45, 12 August 2012 (UTC)

## Something interestingly hexagonal about glimmers and twinkles?

It's supprising but I can't anyone that's managed to pick up on this. When something sparkles such as where a peice of glitter or the surface of water reflects a spot of light into your eye or a camera, especially with out of focus cameras, it always glimers hexagonally. For example go to 50:44 at http://www.bbc.co.uk/iplayer/episode/b0101h6w/Wonders_of_the_Universe_Messengers. Before the camera comes into focus of the stream of water you can initially see a series of sparling white hexagons against a dark background. How come they're always hexagonal and never say triangular or squares? Has anyone looked into this? You would have thought with light sparkling so definely like this it would have been looked into, but strangly it seems unnoticed by everyone. Even with related light patterns that have been actually been used such as the way you often see the sun having a row of cicles lining off from the sun which some game devolopers have taken note of and used in their games haven't seemed to be research into at all, there just 'there'. I can think of a few more examples, but to stay relevent to this article I really think there needs to be some kind of mention and explanation of why light seems to operate in hexagons like this. Robo37 (talk) 17:35, 4 January 2012 (UTC)

Here's another example I've found - http://www.redbubble.com/people/sandra/works/36780-hexagonal-light-refractions-in-rain-drops. Robo37 (talk) 18:35, 4 January 2012 (UTC)

## Imperfect image.

The lines in the following hexagon image are meant to have a slope of ${\displaystyle \scriptstyle {\sqrt {3}}}$ but they instead have a slope of 71/41. The problem is not just in that one image but all over the entire web.

This is how to create a line with that kind of proper exact angle. Blackbombchu (talk) 01:12, 4 August 2013 (UTC)

Images like File:Regular polygon 6.svg are vector plots, computed to at least single-precision. Where does 71/41 come from? Tom Ruen (talk) 02:01, 4 August 2013 (UTC)

I pasted the image into paint then paint poured the outside of the image in red and noticed complete repitition of the edge of the paint poured area every 41 pixels to the right and 71 pixels down. 71^2 actually = 3*41^2 - 2. It should be possible to add a feature to the computer program of automatically generating a line with a slope of ${\displaystyle \scriptstyle {\sqrt {3}}}$ or 1 + ${\displaystyle \scriptstyle {\sqrt {2}}}$ instead of manually trying to get the right slope for the line. Blackbombchu (talk) 03:13, 4 August 2013 (UTC)

If your web-browser supports SVG, the rendering is being done there. If you use Wikipedia's render engine, a 2000 pixel PNG rendering [1], show 790/456 ratio, sqrt(3)*456=789.8152. But whatever numbers you get, its silly to complain about less than 1 pixel error. Tom Ruen (talk) 03:45, 4 August 2013 (UTC)

But 362/209 is a heck of a lot closer; in fact, 362^2 = 3 * 209^2 + 1 and 362 is a smaller number than 790. Blackbombchu (talk) 23:25, 4 August 2013 (UTC)

If you can design your graphics display to support this ratio, then it will display your 362/209. The inaccuracy is a feature of the display (or a particular rendering), not of the image, as explained by Tomruen above. Dbfirs 07:12, 5 August 2013 (UTC)
For fun, here's a 916 pixel rendering [2] that gives your 362:209 ratio, or plus or minus 1 pixel depending on how you interpolate the antialiasing. Tom Ruen (talk) 00:00, 6 August 2013 (UTC)

## hexagon wishing well

I — Preceding unsigned comment added by 173.78.66.181 (talk) 21:35, 11 September 2013 (UTC)

## Symmetries of a regular hexagon

John Conway's book The symmetries of things has a section on symmetry of regular polygons, including a diagram for the hexagon, which I remade below. The original image from the book (p.276) has irregular shaped hexagons which are nice, but I decided to do mine with color markups, and added mirror lines and central rotation orders. I added Coxeter notation symmetry notation, and added v and e subscripts depending if mirrors through vertices or edges. The light purple edges show index 2 subsymmetry, and dark purple index 3. Conway has names for all of them with letters and numbers, the numbers representing the symmetry order. So r2n is (regular) n-gon, dihedral symmetry [n], gn (gyration) is cyclic symmetry, [n]+. dn, pn, and in represent lower mirror symmetries through vertices (diagonal) or edges (perpendicular), or both (isosymmetric) respectively, and a1 for no symmetry. Anyway, I thought I'd put the diagram here for comments. He also has a complete example table for squares (p.272) and a regular octagon. Tom Ruen (talk) 00:00, 7 December 2013 (UTC)